**2. Flow Model Formulations**

A nonregular stretching velocity was used in the flow analysis to characterize horizontal surface movement (for instance, Reference [28]):

$$\mathcal{U}\_{\mathcal{w}}(\mathbf{x},0) = m\mathbf{x}\_{\prime} \tag{1}$$

where *m* is a pilot spreading ratio. The temperature at the surface is *<sup>w</sup>*(*<sup>x</sup>*, 0) = - ∞ + *<sup>m</sup>*<sup>∗</sup>*x*. For the sake of adaptability, it was fabricated as a perpetual at *x* = 0. *<sup>m</sup>\**, *w* and - ∞ provided the temperature variation rate, thermal disparity rate, and ambient temperature congruently.
